Question: Simplify and expand the following expression: $ \dfrac{5p + 9}{4p + 1}+\dfrac{5p}{5p + 4} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4p + 1)(5p + 4)$ Multiply the first term by $\dfrac{5p + 4}{5p + 4}$ $ \begin{align*} \dfrac{5p + 9}{4p + 1} \times \dfrac{5p + 4}{5p + 4} & = \dfrac{(5p + 9)(5p + 4)}{(4p + 1)(5p + 4)} \\ & = \dfrac{25p^2 + 65p + 36}{(4p + 1)(5p + 4)}\end{align*} $ Multiply the second term by $\dfrac{4p + 1}{4p + 1}$ $ \begin{align*} \dfrac{5p}{5p + 4} \times \dfrac{4p + 1}{4p + 1} & = \dfrac{(5p)(4p + 1)}{(5p + 4)(4p + 1)} \\ & = \dfrac{20p^2 + 5p}{(5p + 4)(4p + 1)}\end{align*} $ Now we have: $ = \dfrac{25p^2 + 65p + 36}{(4p + 1)(5p + 4)} + \dfrac{20p^2 + 5p}{(5p + 4)(4p + 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{25p^2 + 65p + 36 + 20p^2 + 5p}{(4p + 1)(5p + 4)} $ $ = \dfrac{45p^2 + 70p + 36}{(4p + 1)(5p + 4)}$ Expand the denominator: $ = \dfrac{45p^2 + 70p + 36}{20p^2 + 21p + 4}$